Soit \( x \) un nombre réel. 1. (a) Montrer que : \( \cos (\arctan (x))=\frac{1}{\sqrt{1+x^{2}}} \). (b) Déduire que \( : \sin (\arctan (x))=\frac{x}{\sqrt{1+x^{2}}} \). 2. (a) Montrer que pour \( x>0: \arctan (x)+\arctan \left(\frac{1}{x}\right)=\frac{\pi}{2} \). (b) Calculer la limite : \( \lim _{x \rightarrow 0^{+}} \frac{1}{x}\left(\arctan \left(\frac{1}{x}\right)-\frac{\pi}{2}\right) \). 3. (a) Montrer que pour tout réel \( t \geq 0: t-\frac{t^{3}}{3} \leq \arctan (t) \leq t \). (b) Calculer la limite : \( \lim _{x \rightarrow+\infty} x\left(1-x \arctan \left(\frac{1}{x}\right)\right) \).

13. Find the derivative of the following function by first expanding the expression. \[ f(x)=(2 x+3)\left(3 x^{2}+1\right) \] \( f^{\prime}(x)=\square \)

a. Use the definition \( m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \) to find the slope of the line tangent to the graph of \( f \) at \( P \). Online Assignment 2-Abdulaziz Jahdali (b. Determine an equation of the tangent line at \( P \). 10. \( f(x)=x^{2}+1, P(-2,5) \) a. \( m_{\tan }=\square \) b. \( y=\square \)

9/16/24, 1:31 \( P M \) 16. Find \( f^{\prime}(x) \) using the rules for finding derivatives, \( f(x)=\frac{6 x-2}{x-2} \) \( f^{\prime}(x)=\square \)

3. Evaluate the following limit. \( \lim _{x \rightarrow-1} \frac{x^{2}-1}{x+1} \) Rewrite the given limit in a way that can be evaluated by direct substitution. \( \lim _{x \rightarrow-1} \frac{x^{2}-1}{x+1}=\lim _{x \rightarrow-1} \) (Simplify your answer.) \( \lim _{x \rightarrow-1} \frac{x^{2}-1}{x+1}=\square \) (Type an exact answer.)

12. Find the derivative of the following function. \[ f(x)=9 x^{5}-29 x+e^{4} \] \( f^{\prime}(x)=\square \)